Homological Serre spectral sequence

Apply the filtered-complex constructions in [F2] to $C_{\ast }(E;R)$ with $F_a{C}_{\ast }=C_{\ast }(E_a;R)$. The resulting differentials have the displayed bidegree and the next page is their homology. By [F1], the initial page vanishes unless $a,b\ge 0$, its first differential is the cellular local boundary, and its next page is $H_a(B;{\mathcal{H}}_b(p;R))$. Since every later term is a subquotient of an earlier one, the first-quadrant vanishing persists.

It remains to prove the finite upper endpoint of the target filtration, which does not follow from chain-level degreewise finiteness. Let $[z]\in H_n(E;R)$ with $n\ge 0$, and write $z$ as a finite cycle. Attach one geometric simplex for each distinct iterated face in its finite support, identifying equally labelled faces. By [F4] this gives a finite CW complex $K$, a map $v:K\to E$, and a cellular $n$-cycle $\tilde{z}$ with $v_{\mathrm{\#}}\tilde{z}=z$. Apply the finite-source clause of [F5] to $pv:K\to B$. It gives a homotopy to a cellular map $g$, so $g(K^j)\subseteq B^j$. Lift this homotopy through $p$, starting at $v$; the endpoint $v^{\mathrm{\prime }}$ satisfies $p{v}^{\mathrm{\prime }}=g$. Every simplex of $\tilde{z}$ has dimension at most $n$, hence $v_{\mathrm{\#}}^{\mathrm{\prime }}\tilde{z}\in C_n(E_n;R)$. The prism identity in [F5] makes it homologous to $z$. Thus every class lies in $F_n{H}_n$, proving $F_n{H}_n=H_n$. Since $F_{-1}C=0$, also $F_{-1}{H}_n=0$. Negative-degree homology is zero.

Suppose now that $B$ is simply connected. For paths $\alpha ,\beta :x\to y$, the loop $\alpha \ast \bar{\beta }$ at $x$ represents the identity by [F6]. Concatenating its endpoint-fixed nullhomotopy with $\beta$ and applying the associativity, inverse, and identity path homotopies gives $[\alpha ]=[\beta ]$ in ${\mathrm{\Pi }}_1(B)$. Thus there is a unique path class between any two points. Functorial fiber transport is consequently path-independent. After fixing one stalk and its unique transport isomorphisms to the other stalks, ${\mathcal{H}}_b$ is the corresponding constant system, giving the untwisted $E^2$ formula.

Here “natural” has the following precise meaning. Given a commutative square of Serre fibrations with total-space map $u:E\to E^{\mathrm{\prime }}$ over a cellular map $f:B\to B^{\mathrm{\prime }}$, [F7] gives $u(E_a)\subseteq E_a^{\mathrm{\prime }}$ and hence a filtered singular-chain map. Its functorial page maps commute with every $d_r$. On $E^1$, the cellwise relative maps commute with the pair connectors, excision maps, and fiber transports used in [F1]; after taking $d_1$-homology this is exactly the $E^2$ map induced by $f$ and the fiber-homology coefficient morphism in [F7]. Thus the displayed spectral sequence is natural for these squares.

Fix a position $(a,b)$ in the first quadrant. The outgoing $d_r$ is zero for $r>a$, because its target has negative first coordinate. The incoming $d_r$ is zero for $r>b+1$, because its source has second coordinate $b-r+1<0$. Hence both incident differentials vanish once $r>\max (a,b+1)$, and the next-page isomorphism in [F2] makes this position stationary. The same bounds show two-sided regularity at every position.

We identify that stationary object. Put $n=a+b$ and $Z_a=F_a{C}_n\cap \ker \mathrm{\partial }$. From the numerator formula in [F2], for $r>a$ the $r$-cycles are exactly $Z_a$, since $F_{a-r}{C}_{n-1}=0$. The first denominator summand is then $F_{a-1}{C}_n\cap \ker \mathrm{\partial }$. Every element of the second summand is an actual boundary lying in $F_a{C}_n$. Conversely, let $x=\mathrm{\partial }y\in F_a{C}_n$. The filtration is exhaustive for this one finite chain, so $y\in F_s{C}_{n+1}$ for some integer $s$. For any $r$ with $a+r-1\ge s$, one has $y\in F_{a+r-1}{C}_{n+1}\cap {\mathrm{\partial }}^{-1}(F_a{C}_n),$ so $x$ belongs to the $r$-boundary denominator. The coherent class of $x$ therefore vanishes on some later page. Once Step 2.1 has reached its stationary range, every transition is an isomorphism, so that class was already zero at the first stationary page. Thus $E_{a,b}^{\mathrm{\infty }}\cong \frac{F_a{C}_n\cap \ker \mathrm{\partial }}{(F_{a-1}{C}_n\cap \ker \mathrm{\partial })+(F_a{C}_n\cap \mathrm{im}\mathrm{\partial })}\mathrm{.}$ This argument chooses a filtration bound only for the displayed $y$; it does not require a uniform bound for all $(n+1)$-chains.

The quotient in Step 3.1 is naturally $F_a{H}_n(E;R)\mathrm{/}{F}_{a-1}{H}_n(E;R)$. Send an actual filtered cycle to its homology class modulo the preceding image filtration. This is surjective by [F3]. If $z\in F_a{C}_n$ maps into $F_{a-1}{H}_n$, there is an actual cycle $w\in F_{a-1}{C}_n$ with $z-w=\mathrm{\partial }y$; hence $z$ lies in the displayed denominator. The converse is immediate. This proves both injectivity and surjectivity and gives the weak-convergence identifications required in [F3].

Step 1.2 makes the target filtration finite, hence exhaustive, separated, and complete by [F3]. Step 2.1 gives two-sided regularity, and Step 4.1 gives the specified weak-convergence isomorphisms. These are exactly the conditions for strong convergence in [F3]. No chain-level assertion $F_n{C}_n=C_n$ was used.

If $B=\mathrm{\varnothing }$, then $E=\mathrm{\varnothing }$ and every page and target is zero; otherwise path connectedness supplies paths used only one at a time. If $E$ or a fiber is empty, the corresponding chains and stalks are zero. The zero ring, $n=0$, $a=0$, $b=0$, the axes, a one-cell finite face complex, and constant or degenerate singular simplices are included in Steps 1.1–4.1. Step 2.1 checks both incoming and outgoing stationary bounds, and Step 4.1 checks both kernel and image directions. Identity and composite naturality follow from [F7]. Every filtration bound, cellular approximation, and lift is attached to one finite representative; no AC or simultaneous choice is used.